Generating Angular Momentum

An object does not just typically have angular momentum. Recall Newton's first law that an object in motion tends to stay in motion. Well, if a figure skater is just skating straight down the ice and then needs to perform a spin or jump with several rotations in the air, he or she needs to generate angular momentum. Angular momentum is generated by the skater applying a force against the ice. The ice then applies a ground reaction force on the skater. This ground reaction force causes gives the skater angular momentum.

The point of application and line of action of this force is critical. If the line of action of the force is directed through the skater's axis of rotation, then he or she won't spin. The force must cause a torque, or moment, which means it must be applied some distance from the axis of rotation AND have a line of action which does not go through the axis of rotation.

The larger the force or the farther the force is from the axis of rotation, the larger the torque. The larger the torque, the greater the angular momentum.

Another key consideration in generating angular momentum is the object's moment of inertia. The larger an object's moment of inertia, the more angular momentum the object can obtain. For example, if a figure skater wants to generate a lot of angular momentum, they should have their arms spread wide, which increases their moment of inertia. In this position, while the skater will have to have a large torque to start rotating, his or angular momentum:

will be larger due to the large I. A skater who starts spinning with his arms at his side, with the same angular velocity will have a smaller angular momentum. Moreover, this skater will not be able to increase his speed in the spin, because he will not be able to reduce his moment of inertia to increase his angular velocity. Two animated figures are provided to illustrate this idea.

The larger the moment of inertia the more torque it takes to start the object spinning. Thus, there is a trade-off between moment of inertia and angular velocity when generating angular momentum. In figure skating, the skaters do not usually have a problem with having producing large enough torque to start spinning. Accordingly, it is to their advantage to start every spin, or rotational trick, with a large moment of inertia. They accomplish this by having their arms and free leg held away from their body.

Some skaters reach rotation speeds of 7 rev/s during a jump. This corresponds to 420 rpm (revolutions per minute). This is as fast than the idling speed of the engine on some cars!!!!

Given the following moments of inertia and angular velocities of the skaters initiating spins, calculate their angular momentum and answer the questions that follow. Note that when calculating angular momentum, it is important to convert any angular velocity to readians/s before performing the calculation.

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